NIH Public Access Author Manuscript Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
NIH-PA Author Manuscript
Published in final edited form as: Conf Proc Int Conf Image Form Xray Comput Tomogr. 2014 ; 2014: 29–32.
Mixed Confidence Estimation for Iterative CT Reconstruction David S. Perlmutter, Soo Mee Kim, Paul E. Kinahan, and Adam M. Alessio Imaging Research Lab, University of Washington, Seattle, Wa USA David S. Perlmutter: [email protected]
Abstract
NIH-PA Author Manuscript
We present a statistical analysis of our previously proposed Constrain-Static Target-Kinetic algorithm for 4D CT reconstruction. This method, where framed iterative reconstruction is only performed on the dynamic regions of each frame, while static regions are fixed across frames to a composite image, was proposed to reduce computation time. In this work, we generalize the previous method to describe any application where a portion of the image is known with higher confidence (static, composite, lower-frequency content, etc.) and a portion of the image is known with lower confidence (dynamic, targeted, etc). We show that by splitting the image space into higher and lower confidence components, CSTK can lower the estimator variance in both regions compared to conventional reconstruction. We present a theoretical argument for this reduction in estimator variance and verify this argument with proof-of-principle simulations. This method allows for reduced computation time and improved image quality for imaging scenarios where portions of the image are known with more certainty than others.
I. Introduction
NIH-PA Author Manuscript
Specific CT imaging applications can have the property that certain regions in the image are known with more confidence than other regions. For example, in dynamic contrast enhanced acquisitions, large regions of the field of view may stay static for each time frame (essentially no change), while targeted regions have varying contrast kinetics. Conventionally, independent reconstructions would be performed for each frame. Several methods have been proposed that use a low-resolution composite image, reconstructed from a time-averaged sino-gram, to aid reconstruction. Composite images have been used as a weight applied to filtered back projection [1] images and as a prior term in a total variation minimization algorithm [2]. We propose a method that uses all frames to reconstruct the static regions and uses individual frames to reconstruct only the kinetic portion (ConstrainStatic Target-Kinetic). This application could be characterized as having side-information to increase the confidence of the static region, while desiring an optimal image of the lowerconfidence, kinetic region. This class of methods is advantageous for both substantially reducing reconstruction time and reducing noise in the lower-confidence region. This general approach could be applied in all applications where side-information could increase the confidence in specific regions, leading to improved image quality in the lowerconfidence area. An obvious application is with multi-frame or gated images where portions of the image do not change between frames (higher confidence regions). Other applications could include images where regions are assumed or known to have lower frequency content than others. For example, when imaging the lungs, it could be assumed that extra-lung
Perlmutter et al.
Page 2
content has much lower spatial frequency and therefore could be known with more confidence than intra-lung content.
NIH-PA Author Manuscript
In our previous work [3] [5], we proposed the Constrain-Static Target-Kinetic (CSTK) reconstruction algorithm as a method to reduce computation time in 4D CT image reconstruction by devoting full computational resources to only the dynamic region of interest. This paper extends that work by presenting an analytic argument, based on an estimator variance analysis [6], that CSTK also improves noise levels throughout the image, including the dynamic region of interest. We feel this analysis can be extended to the situations above, where locally varying performance can be leveraged. We verify our analytic argument with simulation studies. A. CSTK Algorithm
NIH-PA Author Manuscript
Constrain-Static Target-Kinetic reconstruction (CSTK) is a method to reduce computation time of most iterative 4D CT reconstruction algorithms. It comprises the following steps; 1) classify each image pixel as either static or kinetic across frames, perhaps using a high-noise estimate of each frame, 2) form a low-noise, low-resolution “composite image” to initialize all frames, and 3) update only the kinetic pixels in each frame. The resulting computation reduction scales linearly with the percentage of dynamic pixels, minus the time to form the composite image. Previous work [3] showed two applications, Retrospective Gated CT Angiography and Dynamic Perfusion CT, in which CSTK provided similar image quality to conventional OSEM reconstruction with 50% dynamic pixels, and therefore 50% compute time.
II. Statistical Formulation A. Static Model We start by adopting the standard quadratic approximation to the static transmission tomography problem. We wish to estimate the x-ray attenuation coefficients of each pixel in an image, θ = [θ1, …, θm]T ∊ ℝm, from observations Y = [y1, …, yn]T ∊ ℝn, where yi = −log(pi/I0) are the post-log-corrected, measured sinogram bins. By taking the second order Taylor series expansion of the Poisson likelihood [4], the system model can by approximated as,
NIH-PA Author Manuscript
(1)
where A is the tomographic model, a forward projection operator, and Q = diag(1/pi). Recognizing this as a weighted least squares problem, the maximum likelihood estimator θ̂ML can be written explicitly as the least squares solution,
(2)
Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
Perlmutter et al.
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NIH-PA Author Manuscript
̂ is unbiased (E[θ̂ML] = μθ̂ = θ) and that Cov[ θ̂ML] Where W = Q−1. It is easy to show that θML = Σθ̂ = (ATWA)−1. If we choose to add a quadratic prior term to control noise amplification, the solution becomes,
(3)
where R describes the prior term. In the following we will assume no prior for simplicity, although this analysis could be extended to the quadratic prior case. B. Dynamic Model We now consider the 4D extension to the static problem in which we want to estimate multiple images over K time frames, or θ = [θ1, …, θK]T from Y = [Y1, …, YK]T, where each θj ∊ ℝm, Yj ∊ ℝn. A straightforward approach is to treat each frame as a separate static estimation problem, i.e.,
(4a)
NIH-PA Author Manuscript
(4b)
but this solution ignores potentially useful information about θj in Y¬j, where the ¬j superscript indicates all frames besides j. To improve on this, we separate the parameter space into a static and dynamic component, namely, . Previous work on CSTK [3] suggests practical ways to do this partitioning. It is assumed that θs is constant across all frames, and constant only across frame j. This assumption allows us to factor the posterior distribution as,
(5)
NIH-PA Author Manuscript
(6)
The first term in eq. (6) is the conditional of eq. (1), and so is itself multivariate Gaussian variate. The second term is a marginal of the composite image, which is Gaussian distributed if the composite is Gaussian distributed. Although this may not be true for arbitrary composite images, we restrict the composite image to unbiased linear estimators of eq. (1), which we call θĉ with covariance
. Thus, if:
Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
Perlmutter et al.
Page 4
then,
NIH-PA Author Manuscript
(7a)
(7b)
(7c)
(7d)
NIH-PA Author Manuscript
where eq. (7a) come directly from the marginal disribution and eqs. (7b) to (7d) from the conditional distribution of a multivariate Gaussian. Notice that eq. (7b) depends on θs only linearly through its mean. This ensures that the product of eq. (7a) and eq. (7b) is also Gaussian, parameterized by
(8a)
(8b)
Equation (8) is the main result of this paper. Notice that is unbiased, as might be expected (this would not be true if we included a prior term). Also, the covariance can be directly compared with eq. (4), the straightforward reconstruction approach. First, the static pixel variance under CSTK is simply
. A reasonable choice for the composite image
NIH-PA Author Manuscript
would be the average of each frame estimate, θj, in which case . More interestingly, the dynamic pixel variance (lower right term of 8b), is the sum of two terms; and a correction factor, Σ*, where,
To better understand 8b, consider two cases; when and . In the first case, the full dataset Y provides perfect information about θs, and the covariance of the
Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
Perlmutter et al.
Page 5
dynamic portion of the image,
reduces to the conditional covariance
, which is
NIH-PA Author Manuscript
guaranteed to be smaller than . In the second case, the covariance of the static portion of the composite image equals the covariance of the static portion of a single frame; i.e. the composite image did not improve the estimate of θs. Then Σ* = 0, and the covariance is simply . In practice, we expect to lie somewhere in between these two bounds; the composite provides some extra, but not perfect, information about θs, which helps lower the covariance of both dynamic and static pixel estimates. C. Numerical Validation
NIH-PA Author Manuscript
To validate our analysis we compared our predicted covariance with those measured from simulations of a small, 10×10 pixel image. We used a sinogram of 15 detectors × 16 views to make explicit computation of the pseudoinverse tractable. We added Gaussian noise according to the approximate signal model in eq. (1). Figure 1 shows the predicted variance of each pixel using simple framed recon and CSTK, from eq. (4) and eq. (8), respectively. While the true image is static, we nonetheless treat the inner pixels (red pixels in fig. 1, upper left) as dynamic, and collected 10 frames of data. The CSTK image shows dramatic variance reduction in the static region, but also significant reduction in the inner, dynamic region. We then computed θ̂c, , and θ̂CSTK using Y, Y1, and the CSTK method. Figure 1 shows that the predicted, theoretical variance of a single row of pixels agrees well with the sample variance over 1000 noise realizations. Figure 1 also shows the average variance of over all dynamic pixels as the number of frames increases. The framed recon is constant, since it doesn't share information across frames, while the composite image shows the typical
variance reduction. The CSTK variance initially decreases rapidly, then plateaus
to the average variance of
.
III. Experimental Results
NIH-PA Author Manuscript
To demonstrate the utility of these results in a more realistic dynamic CT scenario, we simulated data from a 128×128 dynamic target, pictured in the top row of fig. 4 (the central object is moving continuously at a rate of 8 pixels per frame, where a frame is a full 360 degree revolution of the detector). The dynamic pixels were chosen a priori as an ellipse, pictured in fig. 2. Note, the dynamic pixels are not truly constant within a frame, as assumed in the statistical analysis. The measurement sinogram was 200 detectors × 150 views per frame for 5 frames, with Poisson distributed noise; pi ∼ Poisson(λi), where λi = I0e−Σj ci, j, θj across line of response i. For reconstruction, we use iterative coordinate descent (ICD), but only update dynamic pixels. The choice of composite image is important, and must balance low-noise and computation time. We chose our composite image to be the ICD reconstruction of the average sinogram across all frames. Figure 4 compares the CSTK reconstruction of each frame with that of simple framed recon. The CSTK noise appears lowest, particularly in the static region. However we are primarily concerned with performance in the dynamic region. We present the image roughness (pixel-to-pixel
Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
Perlmutter et al.
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NIH-PA Author Manuscript
variance) in the uniform portion of the dynamic region and total RMSE in the dynamic region. Both metrics are normalized and shown above each image. Figure 3 compares the convergence of both methods. To fairly account for the composite compute time, Tc, in CSTK, the CSTK data is offset by Tc/K, the time to compute reconstruction from averaged sinogram. From fig. 3 and fig. 4, the CSTK method achieves about 15% less noise in the dynamic region in 90% less compute time than simple frame recon. Both factors should increase with number of frames and the percentage of static pixels, as long as the static/ dynamic pixel segmentation remains accurate.
IV. Conclusion and Future Work
NIH-PA Author Manuscript
We have shown that CSTK reconstruction, applied to ML estimation of cardiac gated CT imaging, can both save computation time and lower noise throughout the image. The results confirm the intuition that side information, in the form of increased confidence of particular parameters, can decrease noise. As the strength of the side information increases, i.e. more frames or a higher percentage of static pixels in CSTK and the correlation between side information and pixels of interest increases, the noise reduction is greater. Future work is needed to better understand the effect of CSTK in MAP reconstruction, when a prior term is added to the cost function. We believe mixed confidence estimation can be extended to other applications where local image quality can either benefit from side information or be traded for improved performance elsewhere.
Acknowledgments We thank Drs. Patrick La Riviere and Bruno De Man for helpful conversations. This work is supported by the National Institutes of Health under grants R01-HL109327 and R01-CA115870.
References
NIH-PA Author Manuscript
1. Supanuch M, et al. Radiation dose reduction in time-resolved ct angiography using highly constrained back projection reconstruction. Phys Med Biol. Jul; 2009 54(14):4575–93. [PubMed: 19567941] 2. Chen GH, Tang J, Leng S. Prior image constrained compressed sensing (piccs): a method to accurately reconstruct dynamic ct images from highly undersampled projection data sets. Med Phys. Feb; 2008 35(2):660–3. [PubMed: 18383687] 3. Alessio A, La Riviere P. Constrain static target kinetic iterative image reconstruction for 4d cardiac ct imaging. Proc SPIE. 2011; 7873:78730S–78730S–7. 4. Sauer K, Bouman C. A local update strategy for iterative reconstruction from projections. Signal Processing, IEEE Transactions on. Feb; 1993 41(2):534–548. 5. Alessio A, Kinahan P. Statistical reconstruction of targeted regions with application to ct and pet cardiac imaging. American Journal of Roentgenology (supplement). 2006; 186(4):A46. 6. Fessler J. Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography. Image Processing, IEEE Transactions on. Mar; 1996 5(3): 493–506.
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NIH-PA Author Manuscript NIH-PA Author Manuscript
Fig. 1.
First row presents theoretical variance image for single frame reconstruction and CSTK. The CSTK estimate achieves lower variance everywhere, including the central, dynamic region (yellow region, in upper right). Lower left: Profile through central row of variance image for 3 reconstruction methods, with simulated values as datum and theoretical values as lines. Lower right: Average pixel variance of the dynamic region as a function of frames acquired. Simulated data are datum and theoretical values are lines.
NIH-PA Author Manuscript Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
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NIH-PA Author Manuscript Fig. 2.
Left: Composite image using ICD on the full dataset, Y. Right: Mask for elliptical region of dynamic pixels.
NIH-PA Author Manuscript NIH-PA Author Manuscript Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
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Fig. 3.
Convergence metrics for reconstruction of first frame. Each data point is a full iteration of ICD. Top: Negative log-likelihood value vs. computation time for simple framed (blue) and CSTK (red) recon. Bottom: Mean percent pixel change vs. computation time. Algorithm was terminated when mean pixel change was < 1%.
NIH-PA Author Manuscript Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
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Fig. 4.
Top row: Truth image in the center of each time frame. Row 2-3: Framed ML, and CSTK estimate for each frame. Above each image are image quality metrics of dynamic pixels only. RMSE is the total root mean squared error of the reconstruction. ‘rough’ is the standard deviation of pixels in a flat patch in the dynamic region. Both metrics are normalized.
NIH-PA Author Manuscript Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
NIH-PA Author Manuscript
Published in final edited form as: Conf Proc Int Conf Image Form Xray Comput Tomogr. 2014 ; 2014: 29–32.
Mixed Confidence Estimation for Iterative CT Reconstruction David S. Perlmutter, Soo Mee Kim, Paul E. Kinahan, and Adam M. Alessio Imaging Research Lab, University of Washington, Seattle, Wa USA David S. Perlmutter: [email protected]
Abstract
NIH-PA Author Manuscript
We present a statistical analysis of our previously proposed Constrain-Static Target-Kinetic algorithm for 4D CT reconstruction. This method, where framed iterative reconstruction is only performed on the dynamic regions of each frame, while static regions are fixed across frames to a composite image, was proposed to reduce computation time. In this work, we generalize the previous method to describe any application where a portion of the image is known with higher confidence (static, composite, lower-frequency content, etc.) and a portion of the image is known with lower confidence (dynamic, targeted, etc). We show that by splitting the image space into higher and lower confidence components, CSTK can lower the estimator variance in both regions compared to conventional reconstruction. We present a theoretical argument for this reduction in estimator variance and verify this argument with proof-of-principle simulations. This method allows for reduced computation time and improved image quality for imaging scenarios where portions of the image are known with more certainty than others.
I. Introduction
NIH-PA Author Manuscript
Specific CT imaging applications can have the property that certain regions in the image are known with more confidence than other regions. For example, in dynamic contrast enhanced acquisitions, large regions of the field of view may stay static for each time frame (essentially no change), while targeted regions have varying contrast kinetics. Conventionally, independent reconstructions would be performed for each frame. Several methods have been proposed that use a low-resolution composite image, reconstructed from a time-averaged sino-gram, to aid reconstruction. Composite images have been used as a weight applied to filtered back projection [1] images and as a prior term in a total variation minimization algorithm [2]. We propose a method that uses all frames to reconstruct the static regions and uses individual frames to reconstruct only the kinetic portion (ConstrainStatic Target-Kinetic). This application could be characterized as having side-information to increase the confidence of the static region, while desiring an optimal image of the lowerconfidence, kinetic region. This class of methods is advantageous for both substantially reducing reconstruction time and reducing noise in the lower-confidence region. This general approach could be applied in all applications where side-information could increase the confidence in specific regions, leading to improved image quality in the lowerconfidence area. An obvious application is with multi-frame or gated images where portions of the image do not change between frames (higher confidence regions). Other applications could include images where regions are assumed or known to have lower frequency content than others. For example, when imaging the lungs, it could be assumed that extra-lung
Perlmutter et al.
Page 2
content has much lower spatial frequency and therefore could be known with more confidence than intra-lung content.
NIH-PA Author Manuscript
In our previous work [3] [5], we proposed the Constrain-Static Target-Kinetic (CSTK) reconstruction algorithm as a method to reduce computation time in 4D CT image reconstruction by devoting full computational resources to only the dynamic region of interest. This paper extends that work by presenting an analytic argument, based on an estimator variance analysis [6], that CSTK also improves noise levels throughout the image, including the dynamic region of interest. We feel this analysis can be extended to the situations above, where locally varying performance can be leveraged. We verify our analytic argument with simulation studies. A. CSTK Algorithm
NIH-PA Author Manuscript
Constrain-Static Target-Kinetic reconstruction (CSTK) is a method to reduce computation time of most iterative 4D CT reconstruction algorithms. It comprises the following steps; 1) classify each image pixel as either static or kinetic across frames, perhaps using a high-noise estimate of each frame, 2) form a low-noise, low-resolution “composite image” to initialize all frames, and 3) update only the kinetic pixels in each frame. The resulting computation reduction scales linearly with the percentage of dynamic pixels, minus the time to form the composite image. Previous work [3] showed two applications, Retrospective Gated CT Angiography and Dynamic Perfusion CT, in which CSTK provided similar image quality to conventional OSEM reconstruction with 50% dynamic pixels, and therefore 50% compute time.
II. Statistical Formulation A. Static Model We start by adopting the standard quadratic approximation to the static transmission tomography problem. We wish to estimate the x-ray attenuation coefficients of each pixel in an image, θ = [θ1, …, θm]T ∊ ℝm, from observations Y = [y1, …, yn]T ∊ ℝn, where yi = −log(pi/I0) are the post-log-corrected, measured sinogram bins. By taking the second order Taylor series expansion of the Poisson likelihood [4], the system model can by approximated as,
NIH-PA Author Manuscript
(1)
where A is the tomographic model, a forward projection operator, and Q = diag(1/pi). Recognizing this as a weighted least squares problem, the maximum likelihood estimator θ̂ML can be written explicitly as the least squares solution,
(2)
Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
Perlmutter et al.
Page 3
NIH-PA Author Manuscript
̂ is unbiased (E[θ̂ML] = μθ̂ = θ) and that Cov[ θ̂ML] Where W = Q−1. It is easy to show that θML = Σθ̂ = (ATWA)−1. If we choose to add a quadratic prior term to control noise amplification, the solution becomes,
(3)
where R describes the prior term. In the following we will assume no prior for simplicity, although this analysis could be extended to the quadratic prior case. B. Dynamic Model We now consider the 4D extension to the static problem in which we want to estimate multiple images over K time frames, or θ = [θ1, …, θK]T from Y = [Y1, …, YK]T, where each θj ∊ ℝm, Yj ∊ ℝn. A straightforward approach is to treat each frame as a separate static estimation problem, i.e.,
(4a)
NIH-PA Author Manuscript
(4b)
but this solution ignores potentially useful information about θj in Y¬j, where the ¬j superscript indicates all frames besides j. To improve on this, we separate the parameter space into a static and dynamic component, namely, . Previous work on CSTK [3] suggests practical ways to do this partitioning. It is assumed that θs is constant across all frames, and constant only across frame j. This assumption allows us to factor the posterior distribution as,
(5)
NIH-PA Author Manuscript
(6)
The first term in eq. (6) is the conditional of eq. (1), and so is itself multivariate Gaussian variate. The second term is a marginal of the composite image, which is Gaussian distributed if the composite is Gaussian distributed. Although this may not be true for arbitrary composite images, we restrict the composite image to unbiased linear estimators of eq. (1), which we call θĉ with covariance
. Thus, if:
Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
Perlmutter et al.
Page 4
then,
NIH-PA Author Manuscript
(7a)
(7b)
(7c)
(7d)
NIH-PA Author Manuscript
where eq. (7a) come directly from the marginal disribution and eqs. (7b) to (7d) from the conditional distribution of a multivariate Gaussian. Notice that eq. (7b) depends on θs only linearly through its mean. This ensures that the product of eq. (7a) and eq. (7b) is also Gaussian, parameterized by
(8a)
(8b)
Equation (8) is the main result of this paper. Notice that is unbiased, as might be expected (this would not be true if we included a prior term). Also, the covariance can be directly compared with eq. (4), the straightforward reconstruction approach. First, the static pixel variance under CSTK is simply
. A reasonable choice for the composite image
NIH-PA Author Manuscript
would be the average of each frame estimate, θj, in which case . More interestingly, the dynamic pixel variance (lower right term of 8b), is the sum of two terms; and a correction factor, Σ*, where,
To better understand 8b, consider two cases; when and . In the first case, the full dataset Y provides perfect information about θs, and the covariance of the
Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
Perlmutter et al.
Page 5
dynamic portion of the image,
reduces to the conditional covariance
, which is
NIH-PA Author Manuscript
guaranteed to be smaller than . In the second case, the covariance of the static portion of the composite image equals the covariance of the static portion of a single frame; i.e. the composite image did not improve the estimate of θs. Then Σ* = 0, and the covariance is simply . In practice, we expect to lie somewhere in between these two bounds; the composite provides some extra, but not perfect, information about θs, which helps lower the covariance of both dynamic and static pixel estimates. C. Numerical Validation
NIH-PA Author Manuscript
To validate our analysis we compared our predicted covariance with those measured from simulations of a small, 10×10 pixel image. We used a sinogram of 15 detectors × 16 views to make explicit computation of the pseudoinverse tractable. We added Gaussian noise according to the approximate signal model in eq. (1). Figure 1 shows the predicted variance of each pixel using simple framed recon and CSTK, from eq. (4) and eq. (8), respectively. While the true image is static, we nonetheless treat the inner pixels (red pixels in fig. 1, upper left) as dynamic, and collected 10 frames of data. The CSTK image shows dramatic variance reduction in the static region, but also significant reduction in the inner, dynamic region. We then computed θ̂c, , and θ̂CSTK using Y, Y1, and the CSTK method. Figure 1 shows that the predicted, theoretical variance of a single row of pixels agrees well with the sample variance over 1000 noise realizations. Figure 1 also shows the average variance of over all dynamic pixels as the number of frames increases. The framed recon is constant, since it doesn't share information across frames, while the composite image shows the typical
variance reduction. The CSTK variance initially decreases rapidly, then plateaus
to the average variance of
.
III. Experimental Results
NIH-PA Author Manuscript
To demonstrate the utility of these results in a more realistic dynamic CT scenario, we simulated data from a 128×128 dynamic target, pictured in the top row of fig. 4 (the central object is moving continuously at a rate of 8 pixels per frame, where a frame is a full 360 degree revolution of the detector). The dynamic pixels were chosen a priori as an ellipse, pictured in fig. 2. Note, the dynamic pixels are not truly constant within a frame, as assumed in the statistical analysis. The measurement sinogram was 200 detectors × 150 views per frame for 5 frames, with Poisson distributed noise; pi ∼ Poisson(λi), where λi = I0e−Σj ci, j, θj across line of response i. For reconstruction, we use iterative coordinate descent (ICD), but only update dynamic pixels. The choice of composite image is important, and must balance low-noise and computation time. We chose our composite image to be the ICD reconstruction of the average sinogram across all frames. Figure 4 compares the CSTK reconstruction of each frame with that of simple framed recon. The CSTK noise appears lowest, particularly in the static region. However we are primarily concerned with performance in the dynamic region. We present the image roughness (pixel-to-pixel
Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
Perlmutter et al.
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NIH-PA Author Manuscript
variance) in the uniform portion of the dynamic region and total RMSE in the dynamic region. Both metrics are normalized and shown above each image. Figure 3 compares the convergence of both methods. To fairly account for the composite compute time, Tc, in CSTK, the CSTK data is offset by Tc/K, the time to compute reconstruction from averaged sinogram. From fig. 3 and fig. 4, the CSTK method achieves about 15% less noise in the dynamic region in 90% less compute time than simple frame recon. Both factors should increase with number of frames and the percentage of static pixels, as long as the static/ dynamic pixel segmentation remains accurate.
IV. Conclusion and Future Work
NIH-PA Author Manuscript
We have shown that CSTK reconstruction, applied to ML estimation of cardiac gated CT imaging, can both save computation time and lower noise throughout the image. The results confirm the intuition that side information, in the form of increased confidence of particular parameters, can decrease noise. As the strength of the side information increases, i.e. more frames or a higher percentage of static pixels in CSTK and the correlation between side information and pixels of interest increases, the noise reduction is greater. Future work is needed to better understand the effect of CSTK in MAP reconstruction, when a prior term is added to the cost function. We believe mixed confidence estimation can be extended to other applications where local image quality can either benefit from side information or be traded for improved performance elsewhere.
Acknowledgments We thank Drs. Patrick La Riviere and Bruno De Man for helpful conversations. This work is supported by the National Institutes of Health under grants R01-HL109327 and R01-CA115870.
References
NIH-PA Author Manuscript
1. Supanuch M, et al. Radiation dose reduction in time-resolved ct angiography using highly constrained back projection reconstruction. Phys Med Biol. Jul; 2009 54(14):4575–93. [PubMed: 19567941] 2. Chen GH, Tang J, Leng S. Prior image constrained compressed sensing (piccs): a method to accurately reconstruct dynamic ct images from highly undersampled projection data sets. Med Phys. Feb; 2008 35(2):660–3. [PubMed: 18383687] 3. Alessio A, La Riviere P. Constrain static target kinetic iterative image reconstruction for 4d cardiac ct imaging. Proc SPIE. 2011; 7873:78730S–78730S–7. 4. Sauer K, Bouman C. A local update strategy for iterative reconstruction from projections. Signal Processing, IEEE Transactions on. Feb; 1993 41(2):534–548. 5. Alessio A, Kinahan P. Statistical reconstruction of targeted regions with application to ct and pet cardiac imaging. American Journal of Roentgenology (supplement). 2006; 186(4):A46. 6. Fessler J. Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): applications to tomography. Image Processing, IEEE Transactions on. Mar; 1996 5(3): 493–506.
Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
Perlmutter et al.
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Fig. 1.
First row presents theoretical variance image for single frame reconstruction and CSTK. The CSTK estimate achieves lower variance everywhere, including the central, dynamic region (yellow region, in upper right). Lower left: Profile through central row of variance image for 3 reconstruction methods, with simulated values as datum and theoretical values as lines. Lower right: Average pixel variance of the dynamic region as a function of frames acquired. Simulated data are datum and theoretical values are lines.
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NIH-PA Author Manuscript Fig. 2.
Left: Composite image using ICD on the full dataset, Y. Right: Mask for elliptical region of dynamic pixels.
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Fig. 3.
Convergence metrics for reconstruction of first frame. Each data point is a full iteration of ICD. Top: Negative log-likelihood value vs. computation time for simple framed (blue) and CSTK (red) recon. Bottom: Mean percent pixel change vs. computation time. Algorithm was terminated when mean pixel change was < 1%.
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Fig. 4.
Top row: Truth image in the center of each time frame. Row 2-3: Framed ML, and CSTK estimate for each frame. Above each image are image quality metrics of dynamic pixels only. RMSE is the total root mean squared error of the reconstruction. ‘rough’ is the standard deviation of pixels in a flat patch in the dynamic region. Both metrics are normalized.
NIH-PA Author Manuscript Conf Proc Int Conf Image Form Xray Comput Tomogr. Author manuscript; available in PMC 2015 January 27.
